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Short Communications

No-arbitrage and equivalent martingale measures: an elementary proof of the Harrison–Pliska theorem

Yu. M. Kabanov^a, D. O. Kramkov<sup>b

a</sup> Central Economics and Mathematics Institute, RAS
^b Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We give a new proof of a key result to the theorem that in the discrete-time stochastic model of a frictionless security market the absence of arbitrage possibilities is equivalent to the existence of a probability measure $Q$ which is absolute continuous with respect to the basic probability measure $P$ with the strictly positive and bounded density and such that all security prices are martingales with respect to $Q$. The proof is elementary in a sense that it does not involve a measurable selection theorem.

Keywords: security market, no-arbitrage, equivalent martingale measure.

Received: 02.07.1993

 English version: Theory of Probability and its Applications, 1994, 39:3, 523–527

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